I like Super Mario All-Stars. Exactly the same as the original NES versions (except for a barely noticeable physics glitch in Super Mario Bros. and Lost Levels), but with 16bit audio and visuals. I had absolutely no issues there. Felt more like just a rerelease. Or as if they were late-gen NES games that also had a SNES release at the same time.

A little bit further... would be Pokémon FireRed and LeafGreen Versions. Very true to the originals, with improved audio and visuals. Also a couple of new features and updated game mechanics. Basically "Do you like Pokémon Red and Blue? Well, here it is as if it were made today". Does not offend fans of the originals. And it is still dripping in nostalgia.

HeartGold and SoulSilver, on the other hand, I did not like. They killed the music. And the semi-3D perspective made it feel just one step too different. I did not get any nostalgia from it, which is very important with a remake I think.

Then there are things like HD and 3D remastering. That works too, I guess. If the original look and feel is maintained. I enjoyed OoT 3D, but the new textures were more cartoony than I would have liked. So, those can be good depending on how they're treated.

True remakes or reimagining is different. And... I haven't really played any except for the Wii version of GoldenEye - but that was more an adaptation of the movie, not a remake of the N64 game. So, that doesn't really count. I am looking forward to Tomb Raider though, but I haven't played any of the old games. So, I have nothing to compare that to when I do play it.